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Flat knot 6.1350

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,0,3,2,3,0,1,1,2,1,0,0,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1350']
Arrow polynomial of the knot is: -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384']
Outer characteristic polynomial of the knot is: t^7+57t^5+50t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1350']
2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 2880*K1**4 - 256*K1**3*K3 - 2912*K1**2*K2**2 - 256*K1**2*K2*K4 + 7232*K1**2*K2 - 192*K1**2*K3**2 - 128*K1**2*K4**2 - 4672*K1**2 - 448*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 4832*K1*K2*K3 + 1312*K1*K3*K4 + 224*K1*K4*K5 - 256*K2**4 - 8*K2**2*K4**2 + 968*K2**2*K4 - 4054*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 1936*K3**2 - 884*K4**2 - 80*K5**2 - 2*K6**2 + 4226
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1350']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72429', 'vk6.72481', 'vk6.72493', 'vk6.72840', 'vk6.72852', 'vk6.72901', 'vk6.72908', 'vk6.74450', 'vk6.74460', 'vk6.75063', 'vk6.75073', 'vk6.76962', 'vk6.77782', 'vk6.77972', 'vk6.79458', 'vk6.79906', 'vk6.79916', 'vk6.80931', 'vk6.87231', 'vk6.89366']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -K is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3U1O4O5U6U4U3O6U2U5
R3 orbit {'O1O2O3U1O4O5U6U4U3O6U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5U1U6U5O4O6U3
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U2O5U1U6U5O4O6U3
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 1 0 3 -2],[ 2 0 2 1 0 2 1],[ 0 -2 0 1 1 3 -2],[-1 -1 -1 0 0 1 -2],[ 0 0 -1 0 0 0 0],[-3 -2 -3 -1 0 0 -3],[ 2 -1 2 2 0 3 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 -1 0 -3 -2 -3],[-1 1 0 0 -1 -1 -2],[ 0 0 0 0 -1 0 0],[ 0 3 1 1 0 -2 -2],[ 2 2 1 0 2 0 1],[ 2 3 2 0 2 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,1,0,3,2,3,0,1,1,2,1,0,0,2,2,-1]
Phi over symmetry [-3,-1,0,0,2,2,1,0,3,2,3,0,1,1,2,1,0,0,2,2,-1]
Phi of -K [-2,-2,0,0,1,3,-1,0,2,2,3,0,2,1,2,-1,0,0,1,3,1]
Phi of K* [-3,-1,0,0,2,2,1,0,3,2,3,0,1,1,2,1,0,0,2,2,-1]
Phi of -K* [-2,-2,0,0,1,3,-1,0,2,2,3,0,2,1,2,-1,0,0,1,3,1]
Symmetry type of based matrix +
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+39t^4+28t^2+4
Outer characteristic polynomial t^7+57t^5+50t^3+8t
Flat arrow polynomial -8*K1**2 - 2*K1*K2 + K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial 192*K1**4*K2 - 2880*K1**4 - 256*K1**3*K3 - 2912*K1**2*K2**2 - 256*K1**2*K2*K4 + 7232*K1**2*K2 - 192*K1**2*K3**2 - 128*K1**2*K4**2 - 4672*K1**2 - 448*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 4832*K1*K2*K3 + 1312*K1*K3*K4 + 224*K1*K4*K5 - 256*K2**4 - 8*K2**2*K4**2 + 968*K2**2*K4 - 4054*K2**2 + 32*K2*K3*K5 + 8*K2*K4*K6 - 1936*K3**2 - 884*K4**2 - 80*K5**2 - 2*K6**2 + 4226
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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